December  2012, 5(4): 743-767. doi: 10.3934/krm.2012.5.743

Global existence in critical spaces for the compressible magnetohydrodynamic equations

1. 

Department of Mathematics and Physics, Xiamen University of Technology, Xiamen, Fujian 361024, China

2. 

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China

Received  January 2012 Revised  March 2012 Published  November 2012

In this paper, we are concerned with the global existence and uniqueness of the strong solutions to the compressible Magnetohydrodynamic equations in $\mathbb{R}^N(N\ge3)$. Under the condition that the initial data are close to an equilibrium state with constant density, temperature and magnetic field, we prove the global existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations.
Citation: Qing Chen, Zhong Tan. Global existence in critical spaces for the compressible magnetohydrodynamic equations. Kinetic & Related Models, 2012, 5 (4) : 743-767. doi: 10.3934/krm.2012.5.743
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations,", Springer Verlag, (2011). Google Scholar

[2]

G. Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data,, J. Differential Equations, 182 (2002), 344. Google Scholar

[3]

G. Q. Chen and D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamics equations,, Z. Angew. Math. Phys., 54 (2003), 608. Google Scholar

[4]

Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible Magnetohydrodynamic equations,, Nonlinear Anal., 72 (2010), 4438. Google Scholar

[5]

Q. Chen and Z. Tan, Cauchy problem for the compressible Magnetohydrodynamic equations,, Preprint., (). Google Scholar

[6]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases,, Comm. Partial Differential Equations, 26 (2001), 1183. Google Scholar

[7]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations,, Invent. Math., 141 (2000), 579. Google Scholar

[8]

R. Danchin, Global existence in critical spaces for flows of compressible viscous and Heat-Conductive gases,, Arch. Ration. Mech. Anal., 160 (2001), 1. Google Scholar

[9]

B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars,, Comm. Math. Phys., 266 (2006), 595. Google Scholar

[10]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I,, Arch. Rational Mech. Anal., 16 (1964), 269. Google Scholar

[11]

E. Feireisl, A. Novotný and H. Petleltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids,, J. Math. Fluid Mech., 3 (2001), 358. Google Scholar

[12]

J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations,, Nonlinear Anal., 69 (2008), 3637. Google Scholar

[13]

J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum,, Nonlinear Anal. Real World Appl., 10 (2009), 392. Google Scholar

[14]

J. F. Gerebeau, C. L. Bris and T. Lelievre, "Mathematical Methods for the Magnetohydrodynamics of Liquid Metals,", Oxford University Press, (2006). Google Scholar

[15]

D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics,, Z. Angew. Math. Phys., 56 (2005), 215. Google Scholar

[16]

X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows,, Comm. Math. Phys., 283 (2008), 253. Google Scholar

[17]

X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows,, Arch. Ration. Mech. Anal., 197 (2010), 203. Google Scholar

[18]

S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics,, Proc. Japan Acad. Ser. A, 58 (1982), 384. Google Scholar

[19]

P. L. Lions, "Mathematical Topics in Fluids Mechanics,", Vol. 2, (1998). Google Scholar

[20]

Ta-tsien Li and Tiehu Qin, "Physics and Partial Differential Equations,", 2nd ed., (2005). Google Scholar

[21]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heatconductive fluids,, Proc. Japan Acad. Ser. A, 55 (1979), 337. Google Scholar

[22]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., 20 (1980), 67. Google Scholar

[23]

R. Moreau, "Magnetohydrodynamics,", Kluwer Academic Publishers, (1990). Google Scholar

[24]

A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow,", Oxford University Press, (2004). Google Scholar

[25]

Z. Tan and Y. J. Wang, Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with coulomb force,, Nonlinear Anal., 71 (2009), 5866. Google Scholar

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations,", Springer Verlag, (2011). Google Scholar

[2]

G. Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data,, J. Differential Equations, 182 (2002), 344. Google Scholar

[3]

G. Q. Chen and D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamics equations,, Z. Angew. Math. Phys., 54 (2003), 608. Google Scholar

[4]

Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible Magnetohydrodynamic equations,, Nonlinear Anal., 72 (2010), 4438. Google Scholar

[5]

Q. Chen and Z. Tan, Cauchy problem for the compressible Magnetohydrodynamic equations,, Preprint., (). Google Scholar

[6]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases,, Comm. Partial Differential Equations, 26 (2001), 1183. Google Scholar

[7]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations,, Invent. Math., 141 (2000), 579. Google Scholar

[8]

R. Danchin, Global existence in critical spaces for flows of compressible viscous and Heat-Conductive gases,, Arch. Ration. Mech. Anal., 160 (2001), 1. Google Scholar

[9]

B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars,, Comm. Math. Phys., 266 (2006), 595. Google Scholar

[10]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I,, Arch. Rational Mech. Anal., 16 (1964), 269. Google Scholar

[11]

E. Feireisl, A. Novotný and H. Petleltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids,, J. Math. Fluid Mech., 3 (2001), 358. Google Scholar

[12]

J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations,, Nonlinear Anal., 69 (2008), 3637. Google Scholar

[13]

J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum,, Nonlinear Anal. Real World Appl., 10 (2009), 392. Google Scholar

[14]

J. F. Gerebeau, C. L. Bris and T. Lelievre, "Mathematical Methods for the Magnetohydrodynamics of Liquid Metals,", Oxford University Press, (2006). Google Scholar

[15]

D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics,, Z. Angew. Math. Phys., 56 (2005), 215. Google Scholar

[16]

X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows,, Comm. Math. Phys., 283 (2008), 253. Google Scholar

[17]

X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows,, Arch. Ration. Mech. Anal., 197 (2010), 203. Google Scholar

[18]

S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics,, Proc. Japan Acad. Ser. A, 58 (1982), 384. Google Scholar

[19]

P. L. Lions, "Mathematical Topics in Fluids Mechanics,", Vol. 2, (1998). Google Scholar

[20]

Ta-tsien Li and Tiehu Qin, "Physics and Partial Differential Equations,", 2nd ed., (2005). Google Scholar

[21]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heatconductive fluids,, Proc. Japan Acad. Ser. A, 55 (1979), 337. Google Scholar

[22]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., 20 (1980), 67. Google Scholar

[23]

R. Moreau, "Magnetohydrodynamics,", Kluwer Academic Publishers, (1990). Google Scholar

[24]

A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow,", Oxford University Press, (2004). Google Scholar

[25]

Z. Tan and Y. J. Wang, Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with coulomb force,, Nonlinear Anal., 71 (2009), 5866. Google Scholar

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